See for example Appendix of [1]} for a derivation. Setting \(x = \frac{1}{r}\) , in \((u,x,\theta , \varphi )\)  coordinates, equation (REF ) turns into:
\(2\partial _u\partial _x\phi +\partial _x(Dr^{-2}\partial _x\phi )+{\Delta }_{\mathbb {S}^2}\phi -(2Mx-2e^2 x^2)\phi =0,\) 
